Question

Prove that sin(Π÷14)sin(3Π÷14)sin(5Π÷14)=2​

Answer 1

Answer:

The correct prove will be:

$sin(\frac{\pi }{14})sin(\frac{3\pi }{14})sin(\frac{5\pi }{14})$ $= \frac{1}{8}$

Step-by-step explanation:

    $sin(\frac{\pi }{14})sin(\frac{3\pi }{14})sin(\frac{5\pi }{14})$

            Multiply and Divide by $2cos(\frac{\pi }{14})$

= $\frac{2sin(\frac{\pi }{14})cos(\frac{\pi }{14})sin(\frac{3\pi }{14})sin(\frac{5\pi }{14})}{2cos(\frac{\pi }{14})}$

⇒ Let $\frac{\pi }{14}$ = Ф  and 7Ф = $\frac{\pi }{2}$

= sin2Ф sin3Ф sin5Ф ÷ 2cosФ

= 1/2(2sin2Фsin5Ф)sin3Ф ÷ 2cosФ

= (cos3Ф - cos7Ф) sin3Ф ÷ 4cosФ

= 1/2(2cos3Ф sin3Ф) ÷ 4cosФ               ∵cos7Ф = 0      

= sin6Ф ÷ 8cosФ

= sin(7Ф - Ф) ÷ 8cosФ                              

= (sin7ФcosФ - cos7ФsinФ) ÷ 8cosФ      ∵sin7Ф = 1  

= cosФ ÷ 8cosФ

= 1/8

Hence, The correct prove will be:

$sin(\frac{\pi }{14})sin(\frac{3\pi }{14})sin(\frac{5\pi }{14})$ $= \frac{1}{8}$

Keywords: prove, sinФ, cosФ

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